A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms

: The Maslov form is a closed form for a Lagrangian submanifold of C m , and it is a conformal form if and only if M satisﬁes the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisﬁes a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, ﬁrst, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.


Introduction
It was proven by V. Borrelli, B.-Y. Chen and J. M. Morvan [1], and independently by A. Ros and F. Urbano [2], that if M is a Lagrangian submanifold, with dim(M) = m, of C m , with mean curvature vector H and scalar curvature τ, then for any tangent vector fields X and Y. Thus, they found a simple relationship between one of the main intrinsic invariants, τ, and the main extrinsic invariant H. It was also proven in [2], that the Maslov form, which is a closed form for a Lagrangian submanifold of C m , is a conformal form if and only if M satisfies (1).
Later, D. E. Blair and A. Carriazo [3] established an analogue inequality for anti-invariant submanifolds in R 2m+1 with its standard Sasakian structure and characterized the equality case with a specific expression of the second fundamental form, similar to Equation (1). In a previous paper [4], we studied the corresponding inequality for slant submanifolds of generalized Sasakian space forms; for any tangent vector field Y on M and for certain function f on M.
In such a case, considering an orthonormal basis {e 1 , . . . , e m } on M, it holds that ∇ e i X = f e i , for i = 1, . . . , m.
Now, we will recall some notions about almost-contact Riemannian geometry. For more details about this subject, we recommend the book [7].
Such a manifold is said to be a contact metric manifold if dη = Φ, where Φ(X, Y) = g(X, φY) is called the fundamental 2-form of M. The almost contact metric structure of M is said to be normal if [φ, φ](X, Y) = −2dη(X, Y)ξ, for any X and Y. A normal contact metric manifold is called a Sasakian manifold. It can be proven that an almost contact metric manifold is Sasakian if an only if for any X and Y on M.
In [8], J.A. Oubiña introduced the notion of a trans-Sasakian manifold. An almost contact metric manifold M is a trans-Sasakian manifold if there exists two functions α and β on M such that for any X and Y on M. If β = 0, M is said to be an α-Sasakian manifold. Sasakian manifolds appear as examples of α-Sasakian manifolds, with α = 1. If α = 0, M is said to be a β-Kenmotsu manifold. Kenmotsu manifolds are particular examples with β = 1. If both α and β vanish, then M is a cosymplectic manifold. In particular, from (4) it is easy to see that the following equation holds for a trans-Sasakian manifold: It was proven by J.C. Marrero that, for dimensions greater or equal than 5, the only existing trans-Sasakian manifolds are α-Sasakian and β-Kenmotsu ones [9]. In [10], P. Alegre, D.E. Blair and A. Carriazo introduced the notion of a generalized Sasakian space form as an almost contact metric manifold ( M, φ, ξ, η, g) whose curvature tensor is given by where f 1 , f 2 and f 3 are differential functions on M.
These manifolds are denoted by M( f 1 , f 2 , f 3 ); generalize the notion of Sasakian space form, M(c), whose curvature tensor satisfies the expression (6), with where c is the constant φ-sectional curvature. Now we recall some general definitions and facts about submanifolds. Let M be a submanifold isometrically immersed in a Riemannian manifold ( M,g). We denote by ∇ the induced Levi-Civita connection on M. Thus, the Gauss and Weingarten formulas are respectively given by for vector fields X and Y tangent to M and a vector field V normal to M, where σ denotes the second fundamental form, A V the shape operator in the direction of V and D the normal connection. The second fundamental form and the shape operator are related by M is called a totally geodesic submanifold if σ vanishes identically. We denote by R and R, the curvature tensors of M and M, respectively. They are related by Gauss and Codazzi's equations R(X, Y; Z, W) = R(X, Y; Z, W) where ( R(X, Y)Z) ⊥ denotes the normal component of R(X, Y)Z and is the derivative of Van der Waerden-Bortolotti.
On the other hand, the mean curvature vector H is defined by and M is said to be minimal if H vanishes identically.
The scalar curvature τ of M at p ∈ M is defined by where K(e i , e j ) denotes the sectional curvature of M associated with the plane section spanned by e i and e j , for any tangent vector fields e i and e j in a local orthonormal frame of M. For a submanifold of an almost contact manifold, we denote φX = TX + NX and φV = tV + nV the tangent and normal part of φX and φV for any X tangent vector field and V normal vector field. If the ambient space is trans-Sasakian, taking the tangent and normal part at (4) we obtain: (∇ X n)V + σ(X, tV) + N A V X = 0.
Now, we recall the definition of slant submanifolds. These submanifolds were defined by B.-Y. Chen in [5] on almost Hermitian geometry. Later, A. Lotta defined slant submanifolds on the almost contact metric setting in [11]: given a submanifold M tangent to ξ, for each nonzero vector X tangent to M at p, such that X is not proportional to ξ p , we denote by θ(X) as the angle between φX and T p M. Then, M is said to be slant if the angle θ(X) is a constant, which is independent of the choice of p ∈ M and X ∈ T p M− < ξ p >. The angle θ of a slant immersion is called the slant angle of the immersion. Invariant and anti-invariant immersions are slant immersions with slant angles θ = 0 and θ = π/2, respectively. A slant immersion, which is neither invariant nor anti-invariant, is called a proper slant immersion. Slant submanifolds of Sasakian manifolds were studied by J.L. Cabrerizo, A. Carriazo, L.M. Fernández and M. Fernández in [12,13].
From now on, we denote by m + 1 = 2n + 1 the dimension of M and 2m + 1 = 4n + 1 the dimension of M. We assume m ≥ 2. Then, for a slant submanifold holds: and because of the dimensions, n 2 V = − cos 2 θV, NtV = − sin 2 θV and TtV + tnV = 0, for any X, Y tangent vector fields and V normal vector field. Given a proper slant submanifold M 2n+1 , with slant angle θ, immersed in an almost contact manifold M 4n+1 , we considered an adapted slant reference, [6]; it was built as follows. Given e 1 a unit tangent vector field, orthogonal to ξ, we took: For k > 1, then proceeding by induction, for each l = 1, . . . , n − 1, we chose a unit tangent vector field e 2l+1 of M, such as e 2l+1 , which is orthogonal to e 1 , e 2 , . . . , e 2l−1 , e 2l , ξ and took: In this way {e 1 , . . . , e m , ξ, e 1 * , . . . , e m * } is an orthonormal reference such that e 1 , . . . , e m belong to the contact distribution, D and e 1 * , . . . , e m * are normal to M. Moreover, it can be directly computed that: Finally, a slant submanifold of an (α, β) trans-Sasakian generalized Sasakian space form [4], if its second fundamental form σ is given by the following expression: They are specially interesting because it was proven in [4] that this expression of the second fundamental form characterizes the equality case of the following inequality involving the squared mean curvature H 2 and the scalar curvature τ:

The Maslov Form
For any submanifold of any almost contact manifold, we consider the Maslov form ω H as the dual form of φH; that is for any X tangent vector field in the submanifold. We can also define a canonical 1-form on M by where ω i * i are the connection forms given by Cartan's structure equations.
We can relate these two forms for certain slant submanifolds. In [12], proper slant submanifolds such as for any tangent vector fields X and Y were studied with: They were called slant Sasakian submanifolds in [6]; however, we can point that they are α-Sasakian manifolds with the induced structure φ = sec θT. That aims us to defined slant trans-Sasakian submanifolds as those verifying: For a slant trans-Sasakian submanifold of a trans-Sasakian manifold the relation between the structure functions is given by sec θα = α and β = β.
From (25) and (12) it is deduced that for any X, Y tangent vector fields.
Then, for such a submanifold, the relation between Θ and the Maslov form is given in the following theorem.
Following the same steps that [5] did for slant submanifolds of an almost Hermitian manifold or [6] for an almost contact manifold, and after a long computation, the differentials of θ and η can be proven. The proof is straightforward so we have omitted it. Lemma 1. Let M m+1 , a proper slant submanifold of a generalized Sasakian space, form M 2m+1 endowed with an (α, β) trans-Sasakian structure, with M tangent to ξ and m ≥ 2. Then, the 1-forms Θ and η satisfy: and where θ is the slant angle of M.

Remark 1.
If the ambient space is a Sasakian space form M 2m+1 (c), the Maslov form is closed if and only if c = −3, as it was proved in [6]. In such a case, it holds f 3 = ξ(β).

Remark 2.
We note that on the opposite that for Lagrangian submanifold of C n , [2], or totally real submanifolds of R 2m+1 , [3], the Maslov is not always closed. That aims us to look for an adapted form that is closed in more cases.

An Adapted Closed Form
As the Maslov form is not always closed for slant submanifolds it is necessary to find a new form related with this Maslov form but including the special slant character of the submanifold.
Both the Maslov form and Θ can be considered forms at M or M. As both η and Θ vanish at TM ⊥ , it is the same defining them on M or M; however, it is not the same considering dη or dη M and dΘ or dΘ M . Although both B.-Y. Chen and A. Carriazo, [5] and [6], studied conditions for dω H and dΘ vanishing at the manifold; their real interest was finding a closed form at the submanifold, not at the manifold. Therefore, we consider the restrictions of Θ and η at the submanifold. From (34) and (35) it is deduced: and So we find that, for obtaining a closed form, the relation between Θ and η is not the given by the Maslov form at (28).
Again, we particularize to α-Sasakian or a β-Kenmotsu manifolds. Firstly, we consider an α-Sasakian manifold. It was proven in [14], that if α = 0 and M( f 1 , f 2 , f 3 ) is connected, then α is constant, and the functions are constant and related by f 1 − α 2 = f 2 = f 3 . We can write: From now on, we suppose M is connected.
Moreover, the field associated to the closed form is so we already have the following theorem.
Note that this result improves the one obtained by A. Carriazo in [6] giving a closed form for a slant submanifold of any Sasakian space form. Proof. If M m+1 is a slant submanifold of M 4m+1 (0, −α 2 , −α 2 ), with an α-Sasakian structure. By Theorem 4 the vector field tH + sin 2 θ m + 1 is closed, and the corresponding form is also closed. Therefore it represents a cohomology class in On the other hand, for a β-Kenmotsu manifold dη = 0 and from Theorem 1, ω H = − sin θ m + 1 Θ.
The following lemma studies when it is a closed form.
Therefore, the Maslov form is closed in M if and only f 2 = 0.
Note, that in such a case f 1 − f 3 + ξ(β) + β 2 = 0 ( [14], Proposition 4.3). Moreover, we observe that, on the opposite that for α-Sasakian manifolds, we cannot find a closed form for a slant submanifold of any generalized Sasakian space form with a β-Kenmotsu structure.
However, for f 2 = 0, we have obtained a closed vector field as follows. Again, we can present a topological obstruction for slant immersions: From now on, we will write tH + aξ and tH + bξ, with a = sin 2 θ m + 1 for the correspondent closed vector fields.

About Conformal Forms for α-Sasakian Space Forms
As we said in the Introduction, for those Lagrangian submanifolds of C m verifying the equality case, the Maslov form, that is closed, is also conformal. Now we study if the closed form presented in the previous section is conformal for those slant submanifolds verifying the equality case at (23).
We are considering a connected manifold, so α, f 1 , f 2 and f 3 are constant functions. We want to compute ∇ X (tH + aξ), for any X tangent vector field. It is a long computation. Firstly, we compute ∇N for later use. Using the expression of the second fundamental form of a *-slant submanifold, (22), and (20) in (13): Proof. Firstly, from Codazzi's equation we will compute D X H, and after, ∇ X (tH + aξ). Writing Codazzi's equation, (9), for a generalized Sasakian space form, for any unit orthogonal X, Y tangent vector fields in the contact distribution, using (3) (6), R(X, Y)Y) ⊥ gives: Then, using (40), we obtain: At this point, we use that, taking into account Corollary 1, tH + aξ is a closed vector field.
Then, using (16) g(∇ X tH, Y) = g(∇ Y tH, X) − 2ag(TY, X), and therefore, (14) gives Now, using (22) carries to Now, for dimensions over or equal than 5, we can consider X orthogonal to Y and TY. Multiplying by NX, Interchanging X and Y at (47), and adding it to the previous equation: where X, Y can be interchanged. Comparing (50) with (54) it is proven that and consequently, by (52), g(D X H, NY) = 0, for each X orthogonal to Y and TY at the contact distribution. It only rests on us to compute g(D X H, NTX) in order to know D X H. Multiplying (46) by NTX we obtain: Therefore, taking an orthogonal basis {e * 1 , ..., e * n } at T ⊥ M, for any X unit tangent field orthogonal to ξ. Finally, for any X at the contact distribution, and any Z tangent vector field, g(∇ X (tH + aξ), Z) = g(∇ X tH, Z) − ag(TX, Z) = g(t∇ X H + A nH X − TA H X, Z) − ag(TX, Z) = −g(D X H, NZ) + g(nH, h(X, Z)) + g(H, h(X, TZ)) − ag(TX, Z) The quid point of the above proof is to deduce, from Codazzi's equation and the expression of the second fundamental form, that g(D X H, NX) is independent of X and also that g(D X H, NY) = 0 for Y orthogonal to X, TX. This is the same sketch than A. Ros and F. Urbano did in [2]. Now, we repeat the same steps in order to obtain ∇ ξ (tH + aξ).

Lemma 5.
Let M be *-slant submanifold of a Sasakian space form M(c); it holds: Proof. Using that, from Corollary 1, tH + aξ is a closed vector field, so using (16), for any X tangent vector field, which finishes the proof. Proof. It is a direct consequence of Lemmas 4 and 5.
So, in general, for a *-slant submanifold of a generalized Sasakian space form, the closed form is not conformal. However, for the corresponding vector field, the covariant derivative with respect to X is in the direction of X, TX, tnH and ξ.

About Conformal Forms for β-Kenmotsu Space Forms
At Section 4 we obtained that, for a β-Kenmotsu generalized Sasakian space form M( f 1 , 0, f 3 ), the vector field tH + sin 2 θ m m + 1 ξ = tH + bξ is always closed. So, the associated form plays the role of the Maslov form for Lagrangian submanifolds of Kaehler manifolds. In this section we study if it is conformal for a *-slant submanifold.
The study is similar to the one made at Section 5, so we omit the proofs.
Again, the quid point of the proof is to deduce, from Codazzi's equation and the expression of the second fundamental form, that g(D X H, NX) is independent of X and that g(D X H, NY) = 0 for X orthogonal to Y. Now, we repeat the same steps in order to obtain ∇ ξ (tH + bξ). Proof. It is a direct consequence from Lemmas 6 and 7.
Again, for a *-slant submanifold of a β-Kenmotsu generalized Sasakian space form, the closed form is not conformal. However, for the corresponding vector field, the covariant derivative with respect to X is in the direction of X, TX, tH, tnH and ξ.